M. Ciavarella

Linear elastic contact of the Weierstrass profile

A contact problem is considered in which an elastic half–plane is pressed against a rigid fractally rough surface, whose profile is defined by a Weierstrass series. It is shown that no applied mean pressure is sufficiently large to ensure full contact and indeed there are not even any contact areas of finite dimension — the contact area consists of a set of fractal character for all values of the geometric and loading parameters. A solution for the partial contact of a sinusoidal surface is used to develop a relation between the contact pressure distribution at scale n − 1 and that at scale n. Recursive numerical integration of this relation yields the contact area as a function of scale. An analytical solution to the same problem appropriate at large n is constructed following a technique due to Archard. This is found to give a very good approximation to the numerical results even at small n, except for cases where the dimensionless applied load is large. The contact area is found to decrease continuously with n, tending to a power–law behaviour at large n which corresponds to a limiting fractal dimension of (2 − D), where D is the fractal dimension of the surface profile. However, it is not a ‘simple’ fractal, in the sense that it deviates from the power–law form at low n, at which there is also a dependence on the applied load. Contact segment lengths become smaller at small scales, but an appropriately normalized size distribution tends to a limiting function at large n.† The authors dedicate this paper to the memory of Dr J. F. Archard, 1918–1989.

The generalized cattaneo partial slip plane contact problem. I : Theory

The Cattaneo problem is considered for a general plane contact between elastically similar materials, i.e. a monotonically increasing tangential load, starting from zero, with normal loading held fixed. Instead of the classical argument on the displacement field in the stick zone of Cattaneo solution, we attack the problem implicitly from the governing integral equations in the stick zones. After discussing and solving the full-stick case, we move to the more realistic (for finite friction) case of partial slip. We show that, upon isolating the effect of full sliding, the equalities and inequalities governing the corrective solution for the corrective shearing tractions in the stick zone are exactly the same as those governing the solution of the normal contact problem with a lower load, but the same rotation as the actual one. This analogy permits us to deduce several general properties, and gives a general procedures for solving partial slip Cattaneo problems as frictionless normal indentation ones. Therefore, the general solutions for single, multiple and periodic contacts is given. A comprehensive set of explicit results is given in the part II of the paper.

The indentation modulus of elastically anisotropic materials for indenters of arbitrary shape

Abstract The contact of an indenter of arbitrary shape on an elastically anisotropic half space is considered. It is demonstrated in a theorem that the solution of the contact problem is the one that maximizes the load on the indenter for a given indentation depth. The theorem can be used to derive the best approximate solution in the Rayleigh–Ritz sense if the contact area is a priori assumed to have a certain shape. This approach is used to analyze the contact of a sphere and an axisymmetric cone on an anisotropic half space. The contact area is assumed to be elliptical, which is exact for the sphere and an approximation for the cone. It is further shown that the contact area is exactly elliptical even for conical indenters when a limited class of Greens functions is considered. If only the first term of the surface Greens function Fourier expansion is retained in the solution of the axisymmetric contact problem, a simpler solution is obtained, referred to as the equivalent isotropic solution. For most anisotropic materials, the contact stiffness determined using this approach is very close to the value obtained for both conical and spherical indenters by means of the theorem. Therefore, it is suggested that the equivalent isotropic solution provides a quick and efficient estimate for quantities such as the elastic compliance or stiffness of the contact. The “equivalent indentation modulus”, which depends on material and orientation, is computed for sapphire and diamond single crystals.

The influence of rounded edges on indentation by a flat punch

Abstract The contact problem and stress state for indentation by a flat punch with rounded edges is studied. For the contact problem itself analytical solutions are obtained for both surface pressure and interior stress fields. Cases of normal indentation and frictional contact, the latter in both sliding or partial slip conditions, are all treated. The transition from the Hertzian configuration to the contact between a nominally flat pad and contacting flat surface is discussed, and it is found that the strength of the contact decays surprisingly slowly. Regarding the von Mises yield parameter, there is a range of configurations for which the strength is actually higher than the Hertzian one, and the strength decays only when the corner radii are very small. The present solution is therefore a realistic alternative to the classical rigid-flat punch idealization, and has particular application to fretting fatigue tests.

Incipient sliding of rough surfaces in contact: a multiscale numerical analysis

In this paper, the Cattaneo theory of frictional contact is extended to elastic half-spaces in contact through rough disordered interfaces. The discrete version of the Cattaneo theorem is provided, and represents the basis of a multiscale numerical contact algorithm. Mathematical surfaces with imposed roughness, as well as experimentally digitised ones, are analysed. By means of a numerical method, the evolution of the contact domain, at different resolution, is investigated. Roughness of the interfaces provides lacunarity of the contact domains, whose fractal dimension is always smaller than 2.0. When a tangential force is applied, the extent of the stick area decreases in the same way as the contact area develops with increasing pressure, and the slip area is found to be proportional to the tangential force, as predicted by Cattaneo theory. The evolution of the shear centroid, as well as the amount of dissipated energy up to full-sliding, are provided. Finally, it is shown that, at a sufficient level of discretization, the distribution of contact pressures is multifractal.

A review of analytical aspects of fretting fatigue, with extension to damage parameters, and application to dovetail joints

Abstract Recent advances by the authors in analytical methods for the analysis of plane fretting fatigue (FF) contact problems are described, and new consequences for FF damage are derived. Constant normal load and oscillating tangential load (the celebrated Cattaneo–Mindlin case) are considered with in-phase oscillating moderate bulk stresses, for an arbitrary spline rotated geometry and, in particular, the flat punch with rounded corners in view of application to the dovetail joints. Extremely simple, new results are found for initiation parameters such as tangential microslip and frictional energy, which have been used under certain conditions as threshold parameters for FF. Finally, it is shown that for an “almost flat” geometry, the surface damage parameters decrease, but the tensile stress concentration increases, although it becomes more localized, suggesting that for cracks eventually initiated, the likelihood of self-arrest is higher than in the equivalent Hertzian case with same loads. This seems to interpret recent experiments, although it is not clear whether the optimal geometry in terms of FF life is the perfectly flat one, or an intermediate one.

The generalized Cattaneo partial slip plane contact problem. II—Examples☆

This second part of the paper uses the method devised in the part I to give explicit solution to several cases of Cattaneos plane contact problem, where a monotonically increasing tangential load, starting from zero, is applied to the bodies in contact, with normal loading held fixed. The method consists in reducing the partial slip problem to a superposition of frictionless normal contact problems, for which several results are available, including some recent cases studied by the author. Therefore, a comprehensive set of results is given for single, multiple and periodical contacts.

Indentation by nominally flat or conical indenters with rounded corners

Abstract Axisymmetric indentation of a flat surface is considered: specifically, the case of flat-ended indenter with rounded edges, and the case of a shallow cone with a rounded tip. Analytical solutions are obtained for the normal and sequential tangential loading, in both full or partial slip conditions (with the Cattaneo fn9 Mindlin approximation) , and for the complete interior stress field in all these conditions. Implications for strength of the contact are discussed with reference to metallic or brittle materials, with the intention to shed more light in particular to the understanding of common fretting fatigue or indentation testings with nominally flat or conical indenters. It is found that the strength of the contact, which is nominally zero for perfectly sharp flat or conical indenters, is well defined even for a small radius of curvature. This is particularly true for the flat indenter, for which the strength is even significantly higher than for the classical Hertzian indenter for a wide range of geometrical and loading conditions, rendering it very attractive for design purposes.

On fatigue limit in the presence of notches: classical vs. recent unified formulations

Abstract Classical formulations for the fatigue strength reduction factor of notched specimen, K f , (such as those by Neuber, Peterson, Heywood) were developed long time ago and have found some success by introducing a material constant (dependent on the tensile strength only) in order to take into account the problem of notch sensitivity. However, being empirical fitting equations, they have serious limitations when their asymptotic behaviour is considered, or when the empirical constants are not directly calibrated with experiments. This is shown in this work by using example data taken from the literature for various steels and alloys, and various notch sizes and shapes. Furthermore, although the material constants can be modified to include fatigue threshold dependence (satisfying the requirements of fracture mechanics), only the Neuber formula has a correct functional form in the entire range of notch sizes and shapes, and indeed appears to be sufficiently conservative in the range of data considered. Improved accuracy is found with a more recent empirical criterion due to Atzori and Lazzarin based on the Smith and Miller classification of notches, and with a new criterion here obtained by making consistent the Atzori and Lazzarin with the Lukas–Klesnil, having a sound interpretation in terms of self-arrested cracks ahead of a rounded notch for which the Creager–Paris stress field is valid. A large number of experimental data are taken from the literature to compare the accuracies of the various criteria.

Effect of tip geometry on local indentation modulus measurement via atomic force acoustic microscopy technique

Atomic force acoustic microscopy (AFAM) is a dynamical AFM-based technique very promising for nondestructive analysis of local elastic properties of materials. AFAM technique represents a powerful investigation tool in order to retrieve quantitative evaluations of the mechanical parameters, even at nanoscale. The quantitative determination of elastic properties by AFAM technique is strongly influenced by a number of experimental parameters that, at present, are not fully under control. One of such issues is that the quantitative evaluation require the knowledge of the tip geometry effectively contacting the surface during the measurements. We present and discuss an experimental approach able to determine, at first, tip geometry from contact stiffness measurements and, on the basis of the achieved information, to measure sample indentation modulus. The reliability and the accuracy of the technique has been successfully tested on samples (Si, GaAs, and InP) with very well known structural and morphological ...